JORDAN TRIPLE ENDOMORPHISMS AND ISOMETRIES OF
SPACES OF POSITIVE DEFINITE MATRICES
LAJOS MOLN
´
AR
Abstract. In this paper we determine the structure of certain algebraic mor-
phisms and isometries of the space P
n
of all n × n complex positive definite
matrices. In the case n ≥ 3 we describe all continuous Jordan triple endomor-
phisms of P
n
which are continuous maps φ : P
n
→ P
n
satisfying
φ(ABA) = φ(A)φ(B)φ(A), A, B ∈ P
n
.
It has recently been discovered that surjective isometries of certain substruc-
tures of groups equipped with metrics which are in a way compatible with the
group operations have algebraic properties that relate them rather closely to
Jordan triple morphisms. This makes us possible to use our structural results
to describe all surjective isometries of P
n
that correspond to any member of
a large class of metrics generalizing the geodesic distance in the natural Rie-
mannian structure on P
n
. Finally, we determine the isometry group of P
n
relative to a very recently introduced metric that originates from the diver-
gence called Stein’s loss.
1. Introduction and statement of the main results
The study of analytical and geometrical properties of spaces of positive definite
matrices plays an important role in several areas of pure and applied mathematics
due to the wide spreading applications. One can get an adequate picture of inves-
tigations in that direction and find a lot of information relating to applications in
the monograph [1] by R. Bhatia.
In the present paper we consider the set P
n
of all n ×n complex positive definite
matrices from certain algebraic and metrical points of view. As for the former one,
P
n
becomes an algebraic structure under the Jordan triple product (A, B) 7→ ABA.
We recall that this product is rather extensively investigated in pure ring theory
and also in the theory of operator algebras and its applications. In the case n ≥ 3,
the structure of all continuous Jordan triple automorphisms of P
n
has been deter-
mined in our paper [20] (see Theorem 1 there). Here we substantially strengthen
that former result and describe all continuous Jordan triple endomorphisms of P
n
(which are continuous maps that simply respect the Jordan triple product but not
necessarily bijective).
There is another algebraic operation on P
n
which is closely related to the Jordan
triple product. This is what we have called in our recent paper [13] inverted Jordan
2010 Mathematics Subject Classification. Primary: 15B48, 15A60. Secondary: 15A86, 47B49.
Key words and phrases. Positive definite matrices, Jordan triple endomorphisms, isometries,
geodesic distance, unitarily invariant norms, symmetric Stein divergence.
The author was supported by the ”Lend¨ulet” Program (LP2012-46/2012) of the Hungarian
Academy of Sciences and by the Hungarian Scientific Research Fund (OTKA) Reg.No. K81166
NK81402.
1
2 LAJOS MOLN
´
AR
triple product and defined as (A, B) 7→ AB
−1
A. The reason to introduce that op-
eration is the following. In [13] we have made attempts to generalize Mazur-Ulam
theorem in a non-commutative context. That famous result states that the sur-
jective isometries (i.e., surjective distance preserving maps) between normed real
linear spaces are automatically affine. Our aim has been to find extensions of this
theorem concerning normed spaces to the setting of fairly general non-commutative
metric groups. In the paper [13] we have presented some results that show that
under certain conditions surjective isometries between metric groups or between
certain substructures of groups equipped with metrics that are in some sense com-
patible with the algebraic structure necessarily posses an algebraic property: they
locally preserve the inverted Jordan triple product. We have then utilized this fact
and determined the isometries of different non-linear structures. For example, in
[14] we have described the surjective isometries of the unitary group over a complex
Hilbert space equipped with the operator norm. This result has been generalized
for the context of C
∗
-algebras in [15]. In [23] we have considered the group of
unitary matrices and determined their surjective isometries under general unitarily
invariant norms and also under a class of recently defined metrics that originate
from certain considerations in quantum information processing.
In the present paper we continue that line of investigations. Namely, making
use of our results on Jordan triple endomorphisms of P
n
we determine the isometry
groups of P
n
relative to a large class of metrics. A particular member of that class
is the geodesic distance corresponding to the most natural Riemannian structure
on P
n
. That metric is studied and applied extensively, see, for example, Chapter 6
in [1]. Another particular metric that we are considering is a brand new distance
measure that comes from the symmetric Stein divergence. In fact, in the recent
paper [26] Sra has proved that the square root of that sort of divergence is a true
metric and presented a number of its nice and interesting properties. Below we
determine all surjective isometries of P
n
relative to that metric.
Before presenting our results, we fix the notation. In what follows we denote
by M
n
the space of all n × n complex matrices. Whenever we mention metrical or
topological properties or notions concerning matrices without specifying the metric,
we always have in mind the usual operator norm (or, in another word, spectral
norm) k.k on M
n
(kAk equals the largest singular value of A). The real linear space
of all self-adjoint (or, in another word, Hermitian) elements of M
n
is denoted by
H
n
and U
n
stands for the group of all unitaries in M
n
.
In the first main result of the paper that follows we give the complete description
of the structure of all continuous Jordan triple endomorphisms of P
n
(n ≥ 3) and
hence we generalize the finite dimensional part of Theorem 1 in [20] significantly.
Theorem 1. Assume n ≥ 3. Let φ : P
n
→ P
n
be a continuous map which is a
Jordan triple endomorphism, i.e., φ is a continuous map which satisfies
φ(ABA) = φ(A)φ(B)φ(A), A, B ∈ P
n
.
Then there exist a unitary matrix U ∈ U
n
, a real number c, a set {P
1
, . . . , P
n
}
of mutually orthogonal rank-one projections in M
n
, and a set {c
1
, . . . , c
n
} of real
numbers such that φ is of one of the following forms:
(e1) φ(A) = (det A)
c
UAU
∗
, A ∈ P
n
;
(e2) φ(A) = (det A)
c
UA
−1
U
∗
, A ∈ P
n
;
(e3) φ(A) = (det A)
c
UA
tr
U
∗
, A ∈ P
n
;
3
(e4) φ(A) = (det A)
c
UA
tr
−1
U
∗
, A ∈ P
n
;
(e5) φ(A) =
P
n
j=1
(det A)
c
j
P
j
, A ∈ P
n
.
Above and throughout the paper projection means self-adjoint idempotent. As
an immediate corollary of our first theorem we have the following result which was
originally obtained in [20, Theorem 1].
Corollary 2. Assume n ≥ 3. Let φ : P
n
→ P
n
be a continuous Jordan triple
automorphism, i.e., a continuous bijective map which satisfies
φ(ABA) = φ(A)φ(B)φ(A), A, B ∈ P
n
.
Then there exist a unitary U ∈ U
n
and a number c 6= −1/n such that φ is of one
of the following forms:
(a1) φ(A) = (det A)
c
UAU
∗
, A ∈ P
n
;
(a2) φ(A) = (det A)
c
UA
−1
U
∗
, A ∈ P
n
;
(a3) φ(A) = (det A)
c
UA
tr
U
∗
, A ∈ P
n
;
(a4) φ(A) = (det A)
c
UA
tr
−1
U
∗
, A ∈ P
n
.
The structure of the morphisms above and results what we are about to obtain
on the way leading to that will be utilized to describe the surjective isometries of
P
n
with respect to members of a large class of metrics.
Particular elements of that class have strong differential geometrical origins and
connections. The set P
n
of positive definite matrices is an open subset of the space
H
n
, hence it is a differentiable manifold which can naturally be equipped with a
Riemannian structure in the following way. For any A ∈ P
n
the tangent space of
P
n
at A can be identified with H
n
on which we define an inner product by
hX, Y i
A
= Tr(A
−1/2
XA
−1
Y A
−1/2
), X, Y ∈ H
n
.
The corresponding norm is given by
kXk
A
= kA
−1/2
XA
−1/2
k
HS
, X ∈ H
n
.
Here k.k
HS
stands for the Hilbert-Schmidt norm (or, in another word, Frobenius
norm) which is defined by kT k
2
HS
= Tr(T
∗
T ), T ∈ M
n
. In that way we obtain a
Riemannian space which has long been studied in the literature for many reasons.
For example, it provides probably the most important example of a manifold of
non-positive curvature. The geometry of this manifold is intimately connected with
some matrix inequalities and hence it has a wide range of applications in matrix
analysis. We also point out its connections to problems relating to matrix means,
a really vivid topic in recent days. As to our present results, it is important to
recall that the geodesic distance δ
R
(A, B) between the points A, B ∈ P
n
in this
Riemannian space is given by
δ
R
(A, B) = klog A
−1/2
BA
−1/2
k
HS
.
For details we refer to, e.g., [2] or Chapter 6 in [1].
Connections between means, geodesics, and inequalities were explored in several
interesting papers by G. Corach and his coauthors. In fact, in the 1990’s they
defined and studied a Finsler structure on the manifold of all invertible positive
elements of a general C
∗
-algebra (see, among others, [6], [7], [8]). In the present
setting of matrices this means the following. At any point A in P
n
, on the tangent
space H
n
they defined the Finsler metric (norm) by
kXk
A
= kA
−1/2
XA
−1/2
k, X ∈ H
n
,
4 LAJOS MOLN
´
AR
where k.k is again the usual operator norm (spectral norm). Among revealing many
interesting and important properties of the so-obtained Finsler space they obtained
that the geodesic distance between A and B is given by klog A
−1/2
BA
−1/2
k. Let us
make a short remark here. It is rather surprising that this last quantity appears in
a different context, too. Namely, klog A
−1/2
BA
−1/2
k equals the distance between
A and B relative to the so-called Thompson metric that was defined as a useful
modification of the Hilbert projective metric in a setting much more general than
that of C
∗
-algebras. We refer to our paper [22] where we have determined the
surjective Thompson isometries of the space of all invertible positive operators on
a complex Hilbert space which result has recently been generalized for the setting
of general C
∗
-algebras in [15].
Above we have mentioned norms on H
n
as a tangent space that correspond either
to the Hilbert-Schmidt norm or to the operator norm. In the paper [10] Fujii has
presented a common generalization of the above two approaches for the setting
of finite dimensional C
∗
-algebras. In the case of the algebra M
n
this means the
following (cf. Section 6.4 in [1]). Consider an arbitrary unitarily invariant norm N
on M
n
and define
N(X)
A
= N(A
−1/2
XA
−1/2
)
for each point A ∈ P
n
and every vector X from H
n
. By Theorem 1 in [10], this
formula determines a Finsler metric on P
n
and Theorem 5 in the same paper tells
that the shortest path length d
N
(A, B) between A, B ∈ P
n
is given by
d
N
(A, B) = N(log A
−1/2
BA
−1/2
).
In our next theorem we determine the surjective isometries of P
n
relative to any
of the metrics d
N
.
Theorem 3. Suppose n ≥ 2. Let N be a unitarily invariant norm on M
n
and
φ : P
n
→ P
n
a surjective isometry relative to the metric d
N
. Assume n ≥ 3 and N
is not a scalar multiple of the Hilbert-Schmidt norm. If n 6= 4, then there exists an
invertible matrix T ∈ M
n
such that φ is of one of the following forms:
(t1) φ(A) = T AT
∗
, A ∈ P
n
;
(t2) φ(A) = T A
−1
T
∗
, A ∈ P
n
;
(t3) φ(A) = T A
tr
T
∗
, A ∈ P
n
;
(t4) φ(A) = T A
tr
−1
T
∗
, A ∈ P
n
.
If n = 4, then beside (t1)-(t4) the following additional possibilities can occur:
(d1) φ(A) = (det A)
−2/n
T AT
∗
, A ∈ P
n
;
(d2) φ(A) = (det A)
2/n
T A
−1
T
∗
, A ∈ P
n
;
(d3) φ(A) = (det A)
−2/n
T A
tr
T
∗
, A ∈ P
n
;
(d4) φ(A) = (det A)
2/n
T A
tr
−1
T
∗
, A ∈ P
n
.
In the case where n ≥ 3 and N is a scalar multiple of the Hilbert-Schmidt norm, φ
is of one of the forms (t1)-(t4), (d1)-(d4). Finally, if n = 2, then φ can necessarily
be written in one of the forms (t1)-(t4).
We note that the remarkable fact that in the case n = 4 we have some special
additional possibilities follows from a beautiful general result describing the linear
isometries of symmetric gauge functions which was obtained by
¯
Dokovi´c, Li and
Rodman in [9].
5
We have already mentioned that positive definite matrices play an important
role in several areas of pure and applied mathematics. In many of the ap-
plications the metrical structure of P
n
is of particular interest. For example,
in optimization problems relating to P
n
measuring distances between elements
is a key task and a very nontrivial one when the distance function must re-
spect a non-Euclidean geometry on P
n
. The already mentioned distance measure
δ
R
(A, B) = klog A
−1/2
BA
−1/2
k
HS
is a particularly important example which is
computationally very demanding and also complicated to use. In order to allay
those difficulties, in the paper [26] (also see [25]), Sra has introduced a new metric
on P
n
which not only respects non-Euclidean geometry but offers faster compu-
tation than the previous one and it is also much less complicated to use. In the
mentioned papers several results have been presented that shed light on the advan-
tages of the new metric and relate it to δ
R
in order to justify it is a good proxy for
δ
R
. Moreover, some experimental results have also been given to demonstrate the
usefulness of the new metric which is defined in the following way.
For any pair A, B ∈ P
n
of positive definite matrices their symmetric Stein diver-
gence is defined by
S(A, B) = log det
A + B
2
−
1
2
log det(AB).
Actually, it is just the Jensen-Shannon symmetrization of the divergence called
Stein’s loss (see the first two sections in [26]). In fact, driven by the computational
concerns related to the use of δ
R
, the measure S has originally been introduced in
[4]. The authors of that work claimed that
δ
S
(A, B) =
p
S(A, B), A, B ∈ P
n
is not a metric while the authors in [3] conjectured that it is. The problem has got
a solution in [26, Theorem 5], where it has been proved that δ
S
is a true metric
on P
n
(also see [25]). Beside presenting several interesting results concerning the
properties of the new metric, in the same paper [26] Sra has initiated the study
of the metric space (P
n
, δ
S
) from further aspects. We aim to contribute to his
program by the following theorem in which we determine the precise structure of
the isometry group of (P
n
, δ
S
). This is the last main result of the present paper.
Theorem 4. Assume n ≥ 2. Let φ : P
n
→ P
n
be a surjective isometry relative to
the metric δ
S
. Then there is an invertible matrix T ∈ M
n
such that φ is of one of
the following forms:
(s1) φ(A) = T AT
∗
, A ∈ P
n
;
(s2) φ(A) = T A
−1
T
∗
, A ∈ P
n
;
(s3) φ(A) = T A
tr
T
∗
, A ∈ P
n
;
(s4) φ(A) = T A
tr
−1
T
∗
, A ∈ P
n
.
2. Proofs
In this section we present the proofs of our main results.
We begin with collecting some elementary algebraic properties of Jordan triple
endomorphisms of P
n
. So let φ : P
n
→ P
n
be such a transformation (a Jordan
triple map in short), i.e., assume that
φ(ABA) = φ(A)φ(B)φ(A), A, B ∈ P
n
.